Gas turbine engines often include a plurality of side-by-side shroud segments disposed circumferentially so as to form a circular shroud encircling the blades of a turbine or compressor rotor. Rectangular inter-segments seals are set in slots that are provided at the abutting ends of adjacent shroud segments so as to minimize leakage of the pressurized gases from the main gas path passing inside the shroud. These seals are also called feather seals or strip seals. The axially-extending slots for the inter-segments seals represent a discontinuity in the thermal conduction path at the ends of the shroud segments, with the inner side of the shroud segments somewhat remote from the cooling effect of the cooling air blown on the outer surface. This may adversely affect shroud segment durability at the ends of the shroud segments, particularly where the temperature of the gases in the main gas path is the hottest.